Integrand size = 80, antiderivative size = 26 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=\left (1-4 \left (\frac {11}{5}-2 x+(1+x) \log \left (\frac {3 (3+x)}{x}\right )\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(26)=52\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=\frac {8}{5} \left (-78 x+40 x^2-x (1+40 x) \log \left (3+\frac {9}{x}\right )+10 (1+x)^2 \log ^2\left (3+\frac {9}{x}\right )-39 \log (x)+39 \log (3+x)\right ) \]
Integrate[(-936 - 1848*x + 2256*x^2 + 640*x^3 + (-480 - 984*x - 2408*x^2 - 640*x^3)*Log[(9 + 3*x)/x] + (480*x + 640*x^2 + 160*x^3)*Log[(9 + 3*x)/x]^ 2)/(15*x + 5*x^2),x]
(8*(-78*x + 40*x^2 - x*(1 + 40*x)*Log[3 + 9/x] + 10*(1 + x)^2*Log[3 + 9/x] ^2 - 39*Log[x] + 39*Log[3 + x]))/5
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {640 x^3+2256 x^2+\left (160 x^3+640 x^2+480 x\right ) \log ^2\left (\frac {3 x+9}{x}\right )+\left (-640 x^3-2408 x^2-984 x-480\right ) \log \left (\frac {3 x+9}{x}\right )-1848 x-936}{5 x^2+15 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {640 x^3+2256 x^2+\left (160 x^3+640 x^2+480 x\right ) \log ^2\left (\frac {3 x+9}{x}\right )+\left (-640 x^3-2408 x^2-984 x-480\right ) \log \left (\frac {3 x+9}{x}\right )-1848 x-936}{x (5 x+15)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 \left (80 x^3+282 x^2-231 x-117\right )}{5 x (x+3)}-\frac {8 \left (80 x^3+301 x^2+123 x+60\right ) \log \left (\frac {9}{x}+3\right )}{5 x (x+3)}+32 (x+1) \log ^2\left (\frac {9}{x}+3\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 32 \int (x+1) \log ^2\left (3+\frac {9}{x}\right )dx-96 \operatorname {PolyLog}\left (2,-\frac {x}{3}\right )+64 x^2+64 x^2 \log (x)-64 x^2 \left (\log (x)+\log \left (\frac {3 (x+3)}{x}\right )-\log (3 x+9)\right )-64 x^2 \log (3 x+9)-\frac {624 x}{5}+16 \log ^2(x)+64 \log ^2(3 (x+3))+\frac {488}{5} x \log (x)-\frac {488}{5} x \left (\log (x)+\log \left (\frac {3 (x+3)}{x}\right )-\log (3 x+9)\right )-128 \log \left (\frac {x}{3}+1\right ) \log (x)-32 \log (9) \log (x)-\frac {312 \log (x)}{5}+\frac {336}{5} \log (x+3)-\frac {488}{5} (x+3) \log (3 (x+3))-32 \log (x) \left (\log (x)+\log \left (\frac {3 (x+3)}{x}\right )-\log (3 x+9)\right )+128 \log (x+3) \left (\log (x)+\log \left (\frac {3 (x+3)}{x}\right )-\log (3 x+9)\right )\) |
Int[(-936 - 1848*x + 2256*x^2 + 640*x^3 + (-480 - 984*x - 2408*x^2 - 640*x ^3)*Log[(9 + 3*x)/x] + (480*x + 640*x^2 + 160*x^3)*Log[(9 + 3*x)/x]^2)/(15 *x + 5*x^2),x]
3.29.24.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(21)=42\).
Time = 0.80 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.50
| method | result | size |
| parallelrisch | \(16 x^{2} \ln \left (\frac {3 x +9}{x}\right )^{2}-64 x^{2} \ln \left (\frac {3 x +9}{x}\right )+32 x \ln \left (\frac {3 x +9}{x}\right )^{2}+\frac {864}{5}+64 x^{2}-\frac {8 x \ln \left (\frac {3 x +9}{x}\right )}{5}+16 \ln \left (\frac {3 x +9}{x}\right )^{2}-\frac {624 x}{5}+\frac {312 \ln \left (\frac {3 x +9}{x}\right )}{5}\) | \(91\) |
| norman | \(\frac {312 \ln \left (\frac {3 x +9}{x}\right )}{5}-\frac {624 x}{5}+64 x^{2}+16 \ln \left (\frac {3 x +9}{x}\right )^{2}-\frac {8 x \ln \left (\frac {3 x +9}{x}\right )}{5}+32 x \ln \left (\frac {3 x +9}{x}\right )^{2}-64 x^{2} \ln \left (\frac {3 x +9}{x}\right )+16 x^{2} \ln \left (\frac {3 x +9}{x}\right )^{2}\) | \(96\) |
int(((160*x^3+640*x^2+480*x)*ln((3*x+9)/x)^2+(-640*x^3-2408*x^2-984*x-480) *ln((3*x+9)/x)+640*x^3+2256*x^2-1848*x-936)/(5*x^2+15*x),x,method=_RETURNV ERBOSE)
16*ln(3*(3+x)/x)^2*x^2-64*ln(3*(3+x)/x)*x^2+32*ln(3*(3+x)/x)^2*x+864/5+64* x^2-8/5*x*ln(3*(3+x)/x)+16*ln(3*(3+x)/x)^2-624/5*x+312/5*ln(3*(3+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=16 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right )^{2} + 64 \, x^{2} - \frac {8}{5} \, {\left (40 \, x^{2} + x - 39\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right ) - \frac {624}{5} \, x \]
integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984 *x-480)*log((3*x+9)/x)+640*x^3+2256*x^2-1848*x-936)/(5*x^2+15*x),x, algori thm=\
16*(x^2 + 2*x + 1)*log(3*(x + 3)/x)^2 + 64*x^2 - 8/5*(40*x^2 + x - 39)*log (3*(x + 3)/x) - 624/5*x
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=64 x^{2} - \frac {624 x}{5} + \left (- 64 x^{2} - \frac {8 x}{5}\right ) \log {\left (\frac {3 x + 9}{x} \right )} + \left (16 x^{2} + 32 x + 16\right ) \log {\left (\frac {3 x + 9}{x} \right )}^{2} - \frac {312 \log {\left (x \right )}}{5} + \frac {312 \log {\left (x + 3 \right )}}{5} \]
integrate(((160*x**3+640*x**2+480*x)*ln((3*x+9)/x)**2+(-640*x**3-2408*x**2 -984*x-480)*ln((3*x+9)/x)+640*x**3+2256*x**2-1848*x-936)/(5*x**2+15*x),x)
64*x**2 - 624*x/5 + (-64*x**2 - 8*x/5)*log((3*x + 9)/x) + (16*x**2 + 32*x + 16)*log((3*x + 9)/x)**2 - 312*log(x)/5 + 312*log(x + 3)/5
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.46 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=16 \, {\left (\log \left (3\right )^{2} - 4 \, \log \left (3\right ) + 4\right )} x^{2} + 16 \, {\left (x^{2} + 2 \, x\right )} \log \left (x + 3\right )^{2} + 16 \, {\left (x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + \frac {8}{5} \, {\left (20 \, \log \left (3\right )^{2} - \log \left (3\right ) - 78\right )} x + \frac {8}{5} \, {\left (20 \, x^{2} {\left (\log \left (3\right ) - 2\right )} + x {\left (40 \, \log \left (3\right ) - 1\right )} - 20 \, {\left (x^{2} + 2 \, x\right )} \log \left (x\right ) + 231\right )} \log \left (x + 3\right ) - 16 \, \log \left (x + 3\right )^{2} - \frac {8}{5} \, {\left (20 \, x^{2} {\left (\log \left (3\right ) - 2\right )} + x {\left (40 \, \log \left (3\right ) - 1\right )}\right )} \log \left (x\right ) + 32 \, \log \left (x + 3\right ) \log \left (x\right ) - 16 \, \log \left (x\right )^{2} + 32 \, {\left (\log \left (x + 3\right ) - \log \left (x\right )\right )} \log \left (\frac {9}{x} + 3\right ) - \frac {1536}{5} \, \log \left (x + 3\right ) - \frac {312}{5} \, \log \left (x\right ) \]
integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984 *x-480)*log((3*x+9)/x)+640*x^3+2256*x^2-1848*x-936)/(5*x^2+15*x),x, algori thm=\
16*(log(3)^2 - 4*log(3) + 4)*x^2 + 16*(x^2 + 2*x)*log(x + 3)^2 + 16*(x^2 + 2*x)*log(x)^2 + 8/5*(20*log(3)^2 - log(3) - 78)*x + 8/5*(20*x^2*(log(3) - 2) + x*(40*log(3) - 1) - 20*(x^2 + 2*x)*log(x) + 231)*log(x + 3) - 16*log (x + 3)^2 - 8/5*(20*x^2*(log(3) - 2) + x*(40*log(3) - 1))*log(x) + 32*log( x + 3)*log(x) - 16*log(x)^2 + 32*(log(x + 3) - log(x))*log(9/x + 3) - 1536 /5*log(x + 3) - 312/5*log(x)
\[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=\int { \frac {8 \, {\left (80 \, x^{3} + 20 \, {\left (x^{3} + 4 \, x^{2} + 3 \, x\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right )^{2} + 282 \, x^{2} - {\left (80 \, x^{3} + 301 \, x^{2} + 123 \, x + 60\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right ) - 231 \, x - 117\right )}}{5 \, {\left (x^{2} + 3 \, x\right )}} \,d x } \]
integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984 *x-480)*log((3*x+9)/x)+640*x^3+2256*x^2-1848*x-936)/(5*x^2+15*x),x, algori thm=\
integrate(8/5*(80*x^3 + 20*(x^3 + 4*x^2 + 3*x)*log(3*(x + 3)/x)^2 + 282*x^ 2 - (80*x^3 + 301*x^2 + 123*x + 60)*log(3*(x + 3)/x) - 231*x - 117)/(x^2 + 3*x), x)
Time = 9.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.46 \[ \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx=\frac {312\,\ln \left (\frac {3\,x+9}{x}\right )}{5}-x\,\left (-32\,{\ln \left (\frac {3\,x+9}{x}\right )}^2+\frac {8\,\ln \left (\frac {3\,x+9}{x}\right )}{5}+\frac {624}{5}\right )+16\,{\ln \left (\frac {3\,x+9}{x}\right )}^2+x^2\,\left (16\,{\ln \left (\frac {3\,x+9}{x}\right )}^2-64\,\ln \left (\frac {3\,x+9}{x}\right )+64\right ) \]
int(-(1848*x - log((3*x + 9)/x)^2*(480*x + 640*x^2 + 160*x^3) - 2256*x^2 - 640*x^3 + log((3*x + 9)/x)*(984*x + 2408*x^2 + 640*x^3 + 480) + 936)/(15* x + 5*x^2),x)